Study of the process e+e− → K+K− in the center-of-mass energy range 1010–1060 MeV with the CMD-3 detector

Citation for this paper:

Kozyrev, E.A., Solodov, E.P., Akhmetshin, A.N., Amirkhanov, A.V., Anisenkov, A.V.,

Aulchenko, V.M., … Yudin, Y.V. (2018). Study of the process e

+

_e

−

_{→ K}

+

_K

−

_{in the}

center-of-mass energy range 1010–1060 MeV with the CMD-3 detector. Physics

Letters B, 779, 64-71. https://doi.org/10.1016/j.physletb.2018.01.079

UVicSPACE: Research & Learning Repository

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Faculty Publications

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Study of the process e

+

_e

−

_{→ K}

+

_K

−

_{in the center-of-mass energy range 1010–1060}

MeV with the CMD-3 detector

E.A. Kozyrev, E.P. Solodov, R.R. Akhmetshin, A.N. Amirkhanov, A.V. Anisenkov,

V.M. Aulchenko, V.S. Banzarov, N.S. Bashtovoy… Yu. V. Yudin

2018

© 2017 The Authors. Published by Elsevier B.V. This is an open access article under

the CC BY-NC-ND license (

http://creativecommons.org/licenses/BY-NC-ND/4.0/

).

This article was originally published at:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Study

of

the

process

e

+

e

−

→

K

+

K

−

in

the

center-of-mass

energy

range

1010–1060 MeV

with

the

CMD-3

detector

E.A. Kozyrev

a

,

b

,

∗

,

E.P. Solodov

a

,

b

,

R.R. Akhmetshin

a

,

A.N. Amirkhanov

a

,

b

,

A.V. Anisenkov

a

,

b

,

V.M. Aulchenko

a

,

b

,

V.S. Banzarov

a

,

N.S. Bashtovoy

a

,

D.E. Berkaev

a

,

b

,

A.E. Bondar

a

,

b

,

A.V. Bragin

a

,

S.I. Eidelman

a

,

b

,

D.A. Epifanov

a

,

b

,

L.B. Epshteyn

a

,

b

,

c

,

A.L. Erofeev

a

,

b

_,

_{G.V. Fedotovich}

a

,

b

_,

_{S.E. Gayazov}

a

,

b

_,

_{A.A. Grebenuk}

a

,

b

_,

_{S.S. Gribanov}

a

,

b

_,

D.N. Grigoriev

a

,

b

,

c

,

F.V. Ignatov

a

,

V.L. Ivanov

a

,

b

,

S.V. Karpov

a

,

A.S. Kasaev

a

,

V.F. Kazanin

a

,

b

,

A.A. Korobov

a

,

b

,

I.A. Koop

a

,

A.N. Kozyrev

a

,

b

,

P.P. Krokovny

a

,

b

,

A.E. Kuzmenko

a

,

b

,

A.S. Kuzmin

a

,

b

,

I.B. Logashenko

a

,

b

,

P.A. Lukin

a

,

b

,

A.P. Lysenko

a

,

K.Yu. Mikhailov

a

,

b

,

V.S. Okhapkin

a

,

E.A. Perevedentsev

a

,

b

,

Yu.N. Pestov

a

,

A.S. Popov

a

,

b

,

G.P. Razuvaev

a

,

b

,

Yu.A. Rogovsky

a

,

b

,

A.A. Ruban

a

,

N.M. Ryskulov

a

,

A.E. Ryzhenenkov

a

,

b

,

V.E. Shebalin

a

,

b

,

D.N. Shemyakin

a

,

b

,

B.A. Shwartz

a

,

b

,

D.B. Shwartz

a

,

b

,

A.L. Sibidanov

d

,

Yu.M. Shatunov

a

,

A.A. Talyshev

a

,

b

_,

_{A.I. Vorobiov}

a

_,

_{Yu.V. Yudin}

a

,

b

a_{BudkerInstituteofNuclearPhysics,SBRAS,Novosibirsk,630090,Russia} b_{NovosibirskStateUniversity,Novosibirsk,630090,Russia}

c_{NovosibirskStateTechnicalUniversity,Novosibirsk,630092,Russia} d_{UniversityofVictoria,Victoria,BritishColumbia,V8W3P6,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10October2017

Receivedinrevisedform23January2018 Accepted24January2018

Availableonline2February2018 Editor:L.Rolandi

Theprocesse+e−_→K+K−hasbeenstudiedusing1.7_×106_{eventsfromadatasamplecorresponding}

toanintegratedluminosityof5.7pb−1collectedwiththeCMD-3detectorinthecenter-of-massenergy range1010–1060MeV.Thecrosssectionismeasuredwithabout2%systematicuncertaintyandisused tocalculatethecontributiontotheanomalousmagneticmomentofthemuonaK+K−

μ = (19.33±0.40)× 10−10_{,andtoobtaintheφ(1020)mesonparameters.Weconsidertherelationshipbetweenthee}+_e−_→

K+K−ande+e−→K_S0K0_L crosssectionsandcompareittothetheoreticalprediction.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Investigation of e+e− annihilation into hadrons at low ener-gies provides unique information about interactions of light quarks. A precise measurement of the e+e−

→

K+K−cross section in the center-of-mass energy range Ec.m.

=

1010–1060 MeV allows to

ob-tain the

φ (

1020

)

meson parameters and to estimate a contribution of other light vector mesons, ρ

(

770

),

ω

(

782

)

, to this process. The e+e−

→

K+K− cross section, particularly in the

φ

meson energy region, is also required for a precise calculation of the hadronic contribution to the muon anomaly, aμ, and the value of the fine structure constant at the Z boson

peak, α

(

MZ

)

[1].

*

Correspondingauthor.

E-mailaddress:e.a.kozyrev@inp.nsk.su(E.A. Kozyrev).

The most precise cross section measurements performed by the CMD-2 [2] and BaBar [3] experiments have tension at the level of more than 5% (about 2.6 standard deviations) in the

φ

meson energy region.

Another motivation for this study arises from the comparison of the charged e+e−

→

K+K− and neutral e+e−

→

K0_SK_L0 final states. A significant deviation of the ratio of the coupling con-stants gφ→K+ K−

g_φ→K0 SK 0L

from a theoretical prediction based on previous experiments (see the discussion in Ref.[4]) requires a new precise measurement of the cross sections.

2. CMD-3detectoranddataset

The Cryogenic Magnetic Detector (CMD-3) is a general-purpose detector installed in one of the two interaction regions of the https://doi.org/10.1016/j.physletb.2018.01.079

0370-2693/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

VEPP-2000 collider [5] and is described elsewhere [6]. A detec-tor tracking system consists of a cylindrical drift chamber (DC) and a double-layer cylindrical multiwire proportional chamber (Z-chamber), both installed inside a thin (0.2 X0) superconducting

solenoid with a 1.3 T field. The DC comprises of 1218 hexago-nal cells and allows to measure charged particle momentum with a 1.5–4.5% accuracy in the 100–1000 MeV/c momentum range. It also provides a measurement of the polar (

θ

) and azimuthal (

φ

) angles with an accuracy of 20 mrad and 3.5–8.0 mrad, re-spectively. Amplitude information from the DC wires is used to measure the ionization losses dE/dx of charged particles with a

σ

dE/dx

/

<

dE

/

dx

>

≈

11–14% accuracy for minimum ionization

par-ticles (m.i.p.). The Z-chamber with cathode strip readout is used to calibrate a DC longitudinal scale.

An electromagnetic calorimeter comprised of a liquid xenon volume of a 5.4 radiation length ( X0) thickness followed by CsI

crystals (8.1 X0) outside of the solenoid in the barrel part and BGO

crystals (14.4 X0) in the end cap parts[7,8]. A flux return yoke of

the detector is surrounded by scintillation counters to veto cosmic events.

The beam energy Ebeam is monitored by using the

back-scat-tering laser light system [9,10], which determines Ec.m. at each

energy point with about 0.06 MeV systematic accuracy.

Candidate events are recorded using signals from two indepen-dent trigger systems. One, a charged trigger, uses information only from DC cells indicating the presence of at least one charged track, while the other, a neutral trigger, requires an energy deposition in the calorimeter above Ebeam/2 or the presence of more than two

clusters above 25 MeV threshold.

To study the detector response for the investigated processes and to obtain the detection efficiency, we have developed a Monte Carlo (MC) simulation of the detector based on the GEANT4 [11] package. Simulated events are subject to all reconstruction and selection procedures. MC includes photon jet radiation by initial electron or positron (ISR) calculated according to Ref.[12].

The measurement of the e+e−

→

K+K− cross section pre-sented here is based on a data sample collected at 24 energy points with a 5.7 pb−1 integrated luminosity (IL) in the energy range Ec.m.

=

1010–1060 MeV in 2012 and 2013.

3. Eventselection

Selection of e+e−

→

K+K− candidates is based on the detec-tion of two collinear tracks satisfying the following criteria:

•

The tracks originate from the beam interaction region within 20 cm along the beam axis (Z-coordinate) and within 1 cm in the transverse direction.

•

The polar and azimuthal collinearity are required to have

θ

= |θ

K+

+ θ

K−

−

π

|

,

φ

= ||φ

K+

− φ

K−

|

−

π

|

<

0

.

45

radi-ans. The distributions of these parameters for data and MC at Ebeam

=

530 MeV are shown in Figs. 1,2, where the MC

sam-ple is normalized to data, and arrows demonstrate the applied requirement. Two additional bumps in the

θ

distribution are caused by a significant contribution of K+K−

γ

events, where

γ

is emitted from the initial state (radiative return to the

φ

resonance).

•

The tracks are required to have an average polar angle in the range 1

< θ

aver

= (θ

K+

+

π

− θ

K−

)/

2

<

π

−

1 radians. The polar

angle distribution is shown in Fig. 3(top) where arrows show the applied restriction. Tracks outside the selected range do not pass all DC layers and are detected less efficiently (see the discussion in Sec.6).

•

Momenta of both tracks are required to be close to each other:

|

p1

−

p2

|/|

p1

+

p2

|

<

0

.

3.

Fig. 1. ThepolarcollinearityθK++ θK−−π fordata(points)andMC(shaded

his-togram)atEbeam=530 MeV.

Fig. 2. Theazimuthalcollinearity|φK+− φK−|−πfordata(points)andMC(shaded

histogram)atEbeam=530 MeV.

Fig. 3. (top)Theaveragepolarangleθaver= (θK++π− θK−)/2 distributionfordata (points)andMC(shaded)atEbeam=509.5 MeV.TheMChistogramisnormalizedto

sixcentralbinsofthedatadistribution.(bottom)Thedata-MCratiobefore(points) andafter(squares)applyingefficiencycorrections(seeSec.6).

•

The average momentum of the two tracks is required to be in a range depending on Ebeam to minimize the

background-to-signal ratio. An example of this restriction for Ebeam

=

530 MeV is shown in Fig. 4by arrows: the loss of signal events is less than 0.2% according to MC.

•

In our energy range kaon ionization losses in the DC are signif-icantly larger than those for m.i.p. due to the low momentum of kaons, p

=

100

÷

200 MeV/c. We require both tracks to have ionization losses above a value, which is obtained by taking into account the average value of dE/dx at the measured kaon momentum and dE/dx resolution. The line in Fig. 4 shows an example of the applied selection. As seen in the figure, among

Fig. 4. TheionizationlossesvsmomentumforpositivetracksfordataatEbeam=

530 MeV.Thelinesshowtheacceptanceforthesignalregion.

Fig. 5. DistributionofaverageZ-coordinatesofselectedtracksatEbeam=505 MeV.

Thelong-dottedlinecorrespondstothesignal,the solidlinetothebackground. Theshadedhistogramshowsthebackgrounddistributionobtainedusingeventsat

Ec.m.=984 MeV.

selected events there are those with ISR photons, which have smaller momentum and therefore larger dE/dx. Such events are also retained for further analysis.

The number of signal events is obtained using a fit of the av-erage Z-coordinate distribution of two selected tracks with signal and background functions shown in Fig. 5. The shape of the signal function is described by a sum of two Gaussian distributions with parameters fixed from the simulation, and with additional Gaus-sian smearing to account for the difference in data-MC detector responses. For the background profile we use a second-order poly-nomial function, which describes well a distribution obtained at the energy Ec.m.

=

984 MeV below the threshold of the K+K−

pro-duction shown in Fig. 5by a shaded histogram. The level of back-ground is estimated to be less than 0.5% for all energy points, except for the lowest energy Ec.m.

=

1010

.

46 MeV, where the

back-ground is about 1.1%. The backback-ground is predominantly caused by the beam-gas interaction and interaction of particles lost from the beam at the vacuum pipe walls.

As a result, we obtain 1705060

±

1306 e+e−

→

K+K− signal events.

4. Detectionefficiency

The detection efficiency, MC, is determined from MC by

divid-ing the number of MC simulated events, after reconstruction and selection described above, to the total number of generated K+K−

Fig. 6. TheEXP-MCratioofthesingle-trackefficienciesforpositive EXP+

+MC (squares) andnegative −EXP

−MC

(circles)kaonsfordatacollectedin2012and2013runs.

pairs. The obtained MC is presented in Table 1 from 44% to 55%

and is primarily determined by the restriction on the kaon polar angles and its decays in flight. Simulation of the ISR spectrum de-pends on the cross section under study and this effect is taken into account by iterations. Influence of final-state radiation of real pho-tons (FSR) on MCis examined by including into the MC generator

the FSR amplitude calculated according to scalar electrodynamics with point-like K mesons[12]. The observed change of MC is less

than 0.1%.

Because of some data-MC inconsistency in the tracking effi-ciency, we introduce a correction equal to the ratio of a single-kaon track efficiency in data and MC, _EXP+(−)

/



_MC+(−). A detection efficiency corrected for detector effects is defined as



det

=



MC



_EXP+



_MC+



_EXP−



_MC−

.

(1)

The collinear configuration of the process and large ionization losses allow estimation of the single-kaon track efficiency in data and MC to be performed by selecting a pure class of “test” events with a detected positive or negative charged kaon, and checking how often we reconstruct the opposite track. The detection effi-ciencies for single positive and negative kaons increase from 80% to 90% in our energy range. The data-MC ratios EXP+

_MC+ and −EXP

−_MC of the single-track efficiencies are shown in Fig. 6 for positive (squares) and negative (circles) charged kaons vs c.m. energy, and are used in Eq. (1) to calculate the detection efficiency for each energy point.

5. Crosssectionofe+e−

→

K+K−

The experimental Born cross section of the process e+e−

→

K+K−has been calculated for each energy point according to the expression:

σ

born

=

Nexp



det

·



trig

·

IL

· (

1

+ δ

rad.

)

· (

1

+ δ

en.spr.

),

(2) where trig is a trigger efficiency, IL is the integrated luminosity,

1

+ δ

en.spr._{represents a correction due to the energy spread of the}

electron–positron beams, and 1

+ δ

rad. _{is the initial-state radiative}

correction. The integrated luminosity IL is determined by the pro-cesses e+e−

→

e+e− and e+e−

→

γ γ

with an about accuracy of about 1% [14,15]. The correction 1

+ δ

rad._{, shown by squares in}

Fig. 7, is calculated using the radiative structure function, known with an accuracy better than 0.1%[13].

Table 1

Thec.m.energyEc.m.,numberofselectedsignaleventsN,uncorrectedandcorrecteddetectionefficienciesMCanddet,radiativecorrectionfactor1+ δrad.,correctionfor

thespreadofcollisionenergy1+ δen.spr._{,integratedluminosityIL,andBorncrosssectionσ fortheprocesse}+_e−_→K+_K−_{.Onlystatisticalerrorsareshown.}

Ec.m., MeV N events MC det 1+ δrad. 1+ δen.spr. IL, nb−1 σ, nb

1010.47±0.01 21351±145 0.439 0.441 0.735 0.993 936.05±1.44 69.87±0.50 1012.96±0.01 26882±164 0.485 0.493 0.728 0.988 485.36±1.04 152.45±1.01 1015.07±0.02 6031±78 0.502 0.510 0.718 0.987 47.91±0.33 341.10±5.11 1016.11±0.01 41260±201 0.510 0.513 0.712 0.978 192.11±0.66 575.08±3.84 1017.15±0.02 176768±421 0.515 0.517 0.706 0.983 478.99±1.04 993.19±5.02 1017.16±0.02 22243±149 0.517 0.524 0.706 0.985 60.15±0.30 984.71±8.89 1018.05±0.03 279733±529 0.521 0.519 0.706 0.993 478.34±1.04 1584.27±11.00 1019.12±0.02 270045±520 0.525 0.524 0.721 1.026 328.62±0.86 2228.59±8.13 1019.21±0.03 44051±209 0.525 0.531 0.724 1.022 52.75±0.34 2230.81±18.14 1019.40±0.04 30539±174 0.526 0.533 0.730 1.024 36.05±0.29 2233.66±22.07 1019.90±0.02 391083±626 0.527 0.527 0.752 1.017 472.34±1.04 2127.07±6.46 1021.22±0.03 134598±365 0.532 0.533 0.829 0.994 228.34±0.72 1325.01±9.01 1021.31±0.01 27717±165 0.531 0.540 0.835 0.993 46.85±0.33 1308.31±12.50 1022.08±0.03 89487±299 0.532 0.530 0.885 0.989 201.62±0.68 933.95±6.81 1022.74±0.03 41756±204 0.534 0.536 0.928 0.988 116.71±0.52 710.23±5.86 1023.26±0.04 19718±140 0.536 0.545 0.961 0.991 62.91±0.38 595.03±6.56 1025.32±0.04 7023±84 0.537 0.538 1.077 0.995 36.32±0.29 334.77±5.55 1027.96±0.02 24236±156 0.540 0.536 1.200 0.997 195.83±0.67 191.64±1.74 1029.09±0.02 5786±76 0.542 0.550 1.244 0.997 52.94±0.35 159.94±2.95 1033.91±0.02 11752±108 0.546 0.535 1.392 0.998 175.55±0.64 89.65±1.24 1040.03±0.05 9143±95 0.551 0.553 1.509 0.999 195.91±0.68 55.87±0.94 1049.86±0.02 14818±122 0.553 0.536 1.604 0.999 499.59±1.09 34.47±0.47 1050.86±0.04 4441±67 0.554 0.559 1.609 0.999 146.31±0.59 33.89±0.84 1059.95±0.02 4594±68 0.553 0.543 1.640 0.999 198.86±0.69 25.93±0.64

Fig. 7. Radiativecorrection1+ δrad. _{(squares,leftscale)andcorrection1+ δ}en.spr. forthespreadofcollisionenergy(points,rightscale).

The electron–positron c.m. energy spread, σEc.m., typically about

300 keV, changes the visible cross section. To take into account this effect we apply the following correction:

1

+ δ

en.spr.

(

Ec.m.

)

=

1

√

2

π σ

Ec.m.

·

(3)

·



dE_c.m.

σ

born

₍

_E c.m.

)(

1

+ δ

rad.

(

Ec.m.

))

e −(E_c.m.−Ec.m.)2 2σ_Ec2 .m.

σ

born

₍

_E c.m.

)(

1

+ δ

rad.

(

Ec.m.

))

,

which depends on the cross section σborn_{, radiative correction}

(

1

+ δ

rad.

₎

_{, and is calculated by iterations in the same way as}

MC

and

(

1

+ δ

rad.

₎

_{. The calculated (1}

_{+ δ}

en.spr._{) value for each energy}

point is shown in Fig. 7by circles (right scale), and has the maxi-mum value of 1

.

026

±

0

.

006 at the peak of the

φ

resonance.

The trigger efficiency, trig, is studied using responses of two

in-dependent charged and neutral triggers, for selected signal events, and is found to be close to 100% for the applied selection.

The resulting cross section is listed for each energy point in Ta-ble 1and shown in Fig. 8. The statistical error includes fluctuations of signal and Bhabha events, used for the luminosity calculation,

Fig. 8. Measurede+e−→K+K−crosssection.Thedotsareexperimentaldata,the curveisthefitdescribedinthetext.

and fluctuations of the uncertainty on the c.m. energy measure-ment,

δE

c.m., calculated as

|

∂σ

born

∂Ec.m.

|

× δ

Ec.m..

6. Systematicuncertainties

The uncertainty on the e+e−

→

K+K− cross section is domi-nated by the accuracy on the determination of the detection effi-ciency det.

The systematic uncertainty of the data-MC ratios in Eq. (1) is estimated by applying different selection requirements on the “test” events and does not exceed 1%. However, for five energy points with Ec.m.

>

1030 MeV the uncertainty reaches 2%.

The data-MC difference in the polar angle distributions of kaons is shown in Fig. 3(bottom) by circles. The observed difference is due to incorrect simulation of detector resolution, angular depen-dence of the track reconstruction and trigger efficiency, and un-certainty on the calibration of the DC longitudinal scale. We tune our simulation to match the detector angular and momentum res-olutions (see Fig. 1), to study angular dependence of the track

Table 2

Summaryofsystematicuncertaintiesonthee+e−→K+K−

crosssectionmeasurement.

Source Uncertainty, %

Signal selection 0.3

Detection efficiency 1.6(2.5)

Radiative correction 0.15(0.80)

Energy spread correction 0.3

Trigger efficiency 0.1

Luminosity 1.0

Total 2.0(2.8)

reconstruction efficiency using a single-track test sample, and the response of two independent triggers as a function of the track po-lar angle. The data-MC ratio of the polar angle distributions after applying corrections is shown in Fig. 3(bottom) by squares.

To estimate the influence of the remaining angular uncertainty on the measured cross section we divide all data into three in-dependent samples with

θ

aver

∈

[0.95 : 1.35], [1.35 : π

−

1.35] and

[

π

−

1.35 : π

−

0.95] radians. By separately calculating all parame-ters in Eq.(2)for three regions and comparing the obtained cross sections we estimate the corresponding uncertainty as the average difference of the samples to be 1%.

To check the quality of the DC scale calibration we extrapolate the reconstructed kaon tracks from DC to ZC and compare it with the position of the ZC response: a possible systematic uncertainty is less than 0.3%.

The total systematic uncertainty in the reconstruction efficiency is estimated as 1.6%, but increased to 2.5% for the five energy points with Ec.m.

>

1030 MeV.

To estimate the uncertainty on the background subtraction pro-cedure we use the data accumulated at the energy point Ec.m.

=

984 MeV below the reaction threshold. Applying our selection cri-teria we obtain the number of background events, N984. Then

esti-mate the number of background events for each energy point using the integrated luminosities I L(Ec.m.

)

as:

Nbkg

(

Ec.m.

)

=

N984

·

IL

(

Ec.m.

)

IL

(

984

)

.

(4)

The difference between the expected number of background events and the one obtained by the approximation of the Z-coordinate distribution (see Sec. 3) gives less than 0.3% uncertainty of the cross section: this value is used as an estimate of the correspond-ing systematic uncertainty.

A significant part of selected signal events includes ISR photons, which should be taken into account in the determination of det

and 1

+ δ

rad._{. The photon spectrum is calculated by a convolution}

of the radiator function [13] and Born cross section σborn

₍

_E c.m.

)

which is known with uncertainties discussed above. By varying

σ

born

₍

_E

c.m.

)

according to its systematic uncertainty and repeating

the calculation of the values of (1

+ δ

rad._{) we estimate the}

un-certainty on the last ones as 0.1% (0.8% for energy points with Ec.m.

>

1030 MeV). These values are added quadratically with the

0.1% theoretical uncertainty of the radiator function.

The systematic uncertainties contributing to the measured cross section are listed in Table 2, and the quadratic sum gives a total systematic uncertainty of 2.0% (2.8% for Ec.m.

>

1030 MeV).

7. Approximationofthee+e−

→

K+K−crosssection

The measured cross section defined by Eq.(2)includes a vac-uum polarization factor, Coulomb interaction between K+K−, and final-state radiation of real photons γF S R. We approximate the

en-ergy dependence of the cross section according to the vector

me-son dominance (VMD) model as a squared sum of the ρ

,

ω

, φ

-like amplitudes[18]:

σ

(

s

)

≡

σ

e+e−→K+K−

(

s

)

=

8

π α

3s5/2p 3 K Z

(

s

)

Z

(

m2_φ

)







gφγgφK K Dφ

(

s

)

+

rρ,ω

× [

gργgρK K Dρ

(

s

)

+

gωγgωK K Dω

(

s

)

] +

Aφ,ρ,ω







2

,

(5)

where s

=

E2c.m., pK is the kaon momentum, Z

(

s

)

=

π α

/β

1

−

exp

(

−

π α

/β)



1

+

α

2 4

β

2



(6) is the Sommerfeld–Gamov–Sakharov factor that can be obtained by solving the Schrödinger equation in a Coulomb potential for a P-wave final state with velocity

β

=



1

−

4m2_K

/s,

and

DV

(s)

is the

inverse propagator of the vector state V:

DV

(

s

)

=

m2V

−

s

−

i

√

s



V

(

s

).

(7)

Here mV and



V are mass and width of the major intermediate

resonances: V

=

ρ

(

770

),

ω

(

782

), φ (

1020

)

. For the energy depen-dence of the

φ

meson width we use



φ

(

s

)

= 

φ

·



BK+K− m2_φFK+K−

(

s

)

sFK+K−

(

m2φ

)

+

B_K0 SK0L m2_φF_K0 SKL0

(

s

)

sF_K0 SK0L

(

m 2 φ

)

+

B_π+_π−_π0

√

sF_π+_π−_π0

(

s

)

mφFπ+π−π0

(

m2_φ

)

+

Bηγ Fηγ

(

s

)

Fηγ

(

m2φ

)



,

where F_KK¯

= (

s/4

−

m2_K

)

3/2

_,

Fηγ

(s)

= (

√

s(1

−

m2_η

/s))

3. For the Fπ+_π−_π0

(s)

calculation the model assuming the

φ

→

ρπ

→

π

+

π

−

π

0 _{decay is used[19]. The magnitudes of}

_

ρ

(s)

and



ω

(s)

are calculated in the same way using the corresponding branching fractions [20]. The coupling constants of the intermediate vector meson V with

initial and final states are given by:

gVγ

=



3m3 V



V ee 4

π α

;

gV K+K−

=



6

π

m2 V



VBV K+K− p3_K

(

mV

)

,

where



V ee and BV K+K− are the electronic width and the decay

branching fraction to a kaon pair. In our approximation we use the PDG values for the mass, total width, and electronic width of the ρ

(

770

)

and ω

(

782

)

:



ρ→ee

=

7

.

04

±

0

.

06 keV,



ω→ee

=

0

.

60

±

0

.

02 keV [20]. For the apriori unknown

couplings

of the ρ

(

770

)

and ω

(

782

)

to the pair of kaons we use the relation

g_ωK+K−

=

gρK+K−

= −

gφK+K−

/

√

2

,

(8)

based on the quark model with “ideal” mixing and exact SU(3) symmetry of u-, d-, and s-quarks [18]. In order to take into ac-count a possible breaking of these assumptions, both gρK+K− and

gωK+K− are multiplied by a common complex constant rρ/ω . The amplitude Aφ,ρ,ω denotes a contribution of the higher vector mesons ω

(

1420

),

ρ

(

1450

)

, ω

(

1650

)

,

φ (

1680

)

and ρ

(

1700

)

to the

φ (

1020

)

mass region. Using BaBar [3]and SND [16] data above

√

s

=

1

.

06 GeV for the process e+e−

→

K+K− we extract a contribution of these states.

We perform a fit to the e+e−

→

K+K− cross section with free mφ

,



φ,



φ→ee

×

Bφ→K+K− (or alternatively Bφ→ee

×

Bφ→K+K−)

Fig. 9. Comparisonofthemeasuredcrosssectionwithotherexperimentaldata.Onlystatisticaluncertaintiesareincludedindata.Thewidthofthebandshowsthesystematic uncertainties.

Table 3

Theparametersobtainedfromafitofthecrosssectioncomparedwithpreviousexperiments.

Parameter CMD-3 Other measurements

mφ, MeV 1019.469±0.006±0.060±0.010 1019.461±0.019 (PDG2016)

φ, MeV 4.249±0.010±0.005±0.010 4.266±0.031 (PDG2016)

φ→eeBφ→K+K−, keV 0.669±0.001±0.022±0.005 0.634±0.008 (BaBar)

Bφ→eeBφ→K+K−,10−5 _{15.789±0.033±0.527±0.120 14.24±0.30 (PDG2016)}

and rρ/ω parameters: the fit yields χ2

/ndf

=

25

/

20 ( P

(

χ

2

)

=

20%). The fit result is shown in Fig. 8. Fig. 9 shows the rela-tive difference between the obtained data and the fitted curve. Only statistical errors are shown and the width of the band cor-responds to the systematic uncertainty on the cross section. In Fig. 9(a) we compare our result with previous Novosibirsk mea-surements[2,16,17]while Fig. 9(b) shows a comparison with the recent BaBar data[3]. The obtained parameters of the

φ

meson in comparison with the values of other measurements are presented in Table 3. The first uncertainties are statistical and the second are systematic, resulting from the 60 keV accuracy in the Ec.m.

measurements and errors listed in Table 2. From the fit we ob-tain Re

(rρ

/ω

)

=

0

.

95

±

0

.

03 with an imaginary part compatible with zero. The contributions of the ρ and ω intermediate states (

σ

(s)

−

σ

(s)

|

rρ,ω=0) and higher excitations (

σ

(s)

−

σ

(s)

|

Aφ ,ρ,ω=0)

are shown in Fig. 8as an inset.

To study model dependence of the results, we perform alter-native fits with the Aφ,ρ,ω

=

0 amplitude in Eq.(5), or with an additional floating phase of the

φ

meson amplitude, or with the form of the inverse propagator DV

(s)

=

m2V

−

s

−

imV



V

(s)

in-stead of Eq.(7). The variations of the

φ

meson parameters in these fits are used as an estimate of the model-dependent uncertainty presented as third errors in Table 3.

As shown in Fig. 9, the obtained results have comparable accu-racy but are not consistent, in general, with previous data.

The difference with the CMD-2 [2] measurement can be ex-plained by the overestimation of the value of the trigger efficiency for slow kaons in the previous data. The positive trigger decision from CMD-2 required the presence of one charged track in DC in coincidence with the corresponding hits in the Z-chamber, and with at least one cluster in the CsI calorimeter with an energy deposition greater than 20 MeV. But slow kaons stop in the first wall of the Z-chamber and only decay or their nuclear interaction products can make hits in the Z-chamber or leave energy in the calorimeter. The trigger efficiency of about 90% was obtained actu-ally by simulation, using recorded information from detector cells. In contrast to CMD-2, the new CMD-3 detector has two in-dependent trigger systems, the Z-chamber is excluded from the

decision, and a charged (total) trigger efficiency is close to 100%. The CMD-3 detector has the same Z-chamber with much more de-tailed information, and by including in our selection requirements of hits in the Z-chamber and the presence of an energy deposition greater than 20 MeV in the barrel calorimeter, we obtain a signif-icantly larger trigger efficiency correction than the value obtained in the CMD-2 analysis[2]. A reanalysis of CMD-2 data is expected.

Our value of



φ→eeBφ→K+K− is larger than the BaBar

re-sult by 1.8 standard deviations while the corresponding value of Bφ→eeBφ→K+K− is larger than the PDG one, predominantly based

on the CMD-2 measurement, by 2.7 standard deviations. The ob-tained values of the

φ

meson mass and width agree with the results of other experiments including our recent study of the pro-cess e+e−

→

K0

SKL0[21].

8. Contributiontoaμ

Using the result for the e+e−

→

K+K− cross section we com-pute the contribution of this channel to the muon anomaly aμ via a dispersion relation in the energy region 2mK

<

Ec.m.

<

1.06 GeV.

According to Ref. [1], for the leading-order hadronic contribution we obtain: aK_μ+K−

=



α

mμ 3

π

2(1.06 GeV) 2

4m2 K ds s2K

(

s

)

×

×

σ

(

e+e−

→

K+K−

)

· |

1

− (

s

)

|

2

σ

0

(

e+e−

→

μ

+

μ

−

)

=

= (

19

.

33

±

0

.

04stat

±

0

.

40syst

±

0

.

04VP

)

×

10−10

,

(9)

where K

(s)

is the kernel function, the factor

|

1

− (

s)

|

2 excludes the effect of leptonic and hadronic vacuum polarization (VP), and

σ

0

(e

+e−

→

μ

+

μ

−

)

=

4π α

2

3s is the Born cross section. The first

uncertainty is statistical, the second one corresponds to the sys-tematic uncertainty of σ

(e

+e−

→

K+K−

)

and the third one is the uncertainty of the VP factor (0.2%[23]). In Eq.(9)we integrate the

cross section which includes FSR using the model obtained in the previous section. Then, in order to avoid a model uncertainty, the difference between values of the experimental cross section and the model used, is integrated using the trapezoidal method.

The value should be compared with the recent result of the BaBar collaboration aK+K−

μ

= (

18

.

64

±

0

.

16stat

±

0

.

13syst

±

0

.

03VP

)

×

10−10_{calculated in the same energy range[3]. The difference}

be-tween the two values is 1

.

6

σ

.

9. Comparisonofe+e−

→

K+K−ande+e−

→

K0_SK_L0processes There is a strong relationship between the processes of electron–positron annihilation into K+K− and K0

SKL0 final states.

The difference between them comes from the kinematic effect of the K± and K0 _{mass difference and the Coulomb interaction}

be-tween K+ and K− mesons (Eq. (6)). At the

φ

peak, the Coulomb factor, Z

(m

2_φ

)

, contributes 4.2% to the total cross section. We cor-rect the e+e−

→

K+K−cross section for the above two effects and calculate the difference with the e+e−

→

K0

SK0L cross section: Dc/n

=

σ

e+e−→K+K−

×

β

3_K0

(

s

)

β

3_K±

(

s

)

×

1 Z

(

s

)

−

(10)

−δ

K0_SK_L0

×

σ

e+e−→K_S0K_L0

,

where the factor

δ

_K0

SK0L is introduced to account for a possible

re-maining systematic uncertainty in two measurements: most of the common uncertainties cancel in the difference. The experimental value of Dc/n is shown in Fig. 10by points with error bars, where

the cross section of the production of neutral kaons is taken from our recent measurement[21]. The shaded area in the figure corre-sponds to the systematic uncertainties.

The deviation of Dc/n from zero mostly comes from the

differ-ent structure of the amplitudes of non-resonant isovector states, dominated by the ρ meson, for the processes with charged and neutral kaons. Indeed, instead of relations in Eq.(8)for the charged final state, the coupling constants of the ω

(

782

)

and ρ

(

770

)

with the K0_SK0_L final state are:

g_ωK0

SKL0

= −

gρK0SK0L

= −

gφK0SKL0

/

√

2

,

(11)

where the ρ-meson term has a different sign. So, the magnitude of Dc/n in Eq.(10)is proportional to _DφgρK K_(s)Dgφ K K

ρ(s), that allows to see

experimentally the ρmeson contribution to K-meson production. We fit Dc/n using Eqs.(5), (10)with two free parameters, rρ/ω and

δ

_K0

SK 0

L, discussed above. The mass, width of the

φ

meson and



φ→eeBφ→K+K− are fixed at the values obtained in Sec. 7, also



φ→eeBφ→K0

SKL0 is fixed at 0.428 keV according to Ref.[21]. The fit

result is shown by a solid line in Fig. 10(a) and, in more detail, in insets to Fig. 10(b, c) and yields:

rρ/ω

=

0

.

954

±

0

.

027

,

δ

_K0

SK0L

=

0

.

9964

±

0

.

0014

,

χ

2

/

ndf

=

22

.

2

/

22

.

We obtain good description of data by the fit. A small deviation of rρ/ω from

unity demonstrates the accuracy (

≈

5%) of relations(8), (11)and confirms that the contribution from the ρ

(

770

)

meson to Dc/n dominates in the energy range under study. The deviation of

δ

_K0

SK0L from unity (0.36%) shows the level of a possible remaining

systematic uncertainty of the cross section measurements.

Additionally, from the comparison of the charged and neutral cross sections we can obtain the ratio of the coupling constants:

Fig. 10. ThedifferenceofthechargedandneutralcrosssectionsdefinedasDc/n=

σe+e−→K+K−× β3 K 0(s) β3 K±(s) × 1 Z(s)− δK0

SK0L×σe+e−→K0SK0L.Theshadedareacorresponds

tosystematicuncertaintiesindata,thesolidlinetothefitdescribedinthetext.

R

=

gφK+K− g_φK0 SK0L



Z

(

m2_φ

)

=







B

(φ

→

K+K−

)

B

(φ

→

K0_SK_L0

)

·

1 Z

(

m2 φ

)

·

β

3 K0

β

3_K±

=

0

.

990

±

0

.

017

,

where the common parts of systematic uncertainties originating from luminosity, radiative and energy spread corrections, are also reduced. As expected from isospin symmetry of u- and d-quarks, the value of R is

consistent with unity.

Additionally to the Coulomb interaction taken into account by the factor Z

(s)

, the final-state radiation of real photons, according to Ref.[22], decreases the total e+e−

→

K+K−

(

γ

)

cross section by about 0.4% at the

φ

meson mass. This effect partially explains the deviation of R from

unity.

10. Conclusion

Using CMD-3 data in the Ec.m.

=

1010–1060 MeV energy range

we select 1

.

7

×

106_{events of the process e}+_e−

_→

_K+_K−_{, and}

mea-sure the cross section with a systematic error of about 2%. By a fit of the cross section in the VMD model the following values of the

φ

meson parameters have been obtained:

mφ

=

1019

.

469

±

0

.

061 MeV

/

c2



φ

=

4

.

249

±

0

.

015 MeV



φ→eeBφ→K+K−

=

0

.

669

±

0

.

023 keV

We calculate the contribution of the obtained cross section to the anomalous magnetic moment of the muon aK_μ+K−

= (

19

.

33

±

0

.

40

)

×

10−10 _{in the energy range from threshold to}

√

_s

₌

1

.

06 GeV.

The observed deviation of the ρ

(

770

)

and ω

(

782

)

amplitudes, rρ/ω

=

0

.

95

±

0

.

03, from a naive theoretical prediction, allows to estimate the accuracy of the used VMD-based phenomenologi-cal model to better than 5%. The obtained ratio gφ K + K −

g φ K 0_SK 0_L  Z(m2 φ)

=

0

.

990

±

0

.

017 is consistent with isospin symmetry.

Acknowledgements

We thank the VEPP-2000 personnel for the excellent machine operation. This work is supported in part by the Russian Educa-tion and Science Ministry, and by the Russian FoundaEduca-tion for Basic Research grants RFBR 15-02-05674, 17-02-00897, 17-52-50064. References

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